Almost all my keyboards are white due to them being vintage, and they attract dirt and dust, especially between the keys. While I have a special plastic cleaner that is alcohol-free and easy to use on most surfaces with a microfiber wipe, I have problem with between the keys
Should I order some specialty kit, or just get some brush from some local hardware store? Makeup brushes don't work very well for this purpose in my experience.
I just use a stiff makeup brush, while blowing... seems to get most of the dust out. Occasionally, I will remove the keycaps and brush it out properly. By far the best way to clean it, is to keep it clean though. Use a dust cover when not in use. Anything will do... even an old t shirt. Another advantage of that is keeping UV off it when not in use, especially with older, vintage boards, as it slows down ageing and yellowing.
Personally I've got a datavac that I use for blowing the dust and fur out of my desktop, and end up using it to blow any dust and dirt that lands between the keys. Of course if you've got some crusty build up that's sat there for long then...you'll probably need to remove the caps, maybe take apart and use some 99% isopropyl alcohol and paper towels or something to really clean it up.
To get out dirt and dust I just use a vacuum cleaner. Remove all the keycaps and vacuum out the case, and a little brush can help get whatever the vacuum doesn't get.